Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle α.

Right scalene triangle.

Sides: a = 82   b = 66.48330805544   c = 48

Area: T = 1595.594393331
Perimeter: p = 196.4833080554
Semiperimeter: s = 98.24215402772

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 54.17111598996° = 54°10'16″ = 0.94554650999 rad
Angle ∠ C = γ = 35.82988401004° = 35°49'44″ = 0.62553312269 rad

Height: ha = 38.91769252026
Height: hb = 48
Height: hc = 66.48330805544

Median: ma = 41
Median: mb = 58.3876642308
Median: mc = 70.68223881883

Inradius: r = 16.24215402772
Circumradius: R = 41

Vertex coordinates: A[48; 0] B[0; 0] C[48; 66.48330805544]
Centroid: CG[32; 22.16110268515]
Coordinates of the circumscribed circle: U[24; 33.24215402772]
Coordinates of the inscribed circle: I[31.75884597228; 16.24215402772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 125.82988401° = 125°49'44″ = 0.94554650999 rad
∠ C' = γ' = 144.17111599° = 144°10'16″ = 0.62553312269 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 82 ; ; c = 48 ; ; alpha = 90° ; ;

2. From angle α, side c and side a we calculate side b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 82**2 = 48**2 + b**2 - 2 * 48 * b * cos 90° ; ; ; ; ; ; b**2 -4420 =0 ; ; a=1; b=-0; c=-4420 ; ; D = b**2 - 4ac = 0**2 - 4 * 1 * (-4420) = 17680 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ ± sqrt{ 17680 } }{ 2 } = fraction{ ± 4 sqrt{ 1105 } }{ 2 } ; ; b_{1,2} = ± 2 sqrt{ 1105} = ± 66.4830805544 ; ; b_{1} = 2 sqrt{ 1105} = 66.4830805544 ; ;
b_{2} = - 2 sqrt{ 1105} = -66.4830805544 ; ; ; ; text{ Factored form: } ; ; (b -66.4830805544) (b +66.4830805544) = 0 ; ; ; ; b > 0 ; ; ; ; b = 66.483 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 82 ; ; b = 66.48 ; ; c = 48 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 82+66.48+48 = 196.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 196.48 }{ 2 } = 98.24 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 98.24 * (98.24-82)(98.24-66.48)(98.24-48) } ; ; T = sqrt{ 2545920 } = 1595.59 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1595.59 }{ 82 } = 38.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1595.59 }{ 66.48 } = 48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1595.59 }{ 48 } = 66.48 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 66.48**2+48**2-82**2 }{ 2 * 66.48 * 48 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 82**2+48**2-66.48**2 }{ 2 * 82 * 48 } ) = 54° 10'16" ; ; gamma = 180° - alpha - beta = 180° - 90° - 54° 10'16" = 35° 49'44" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1595.59 }{ 98.24 } = 16.24 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 82 }{ 2 * sin 90° } = 41 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 66.48**2+2 * 48**2 - 82**2 } }{ 2 } = 41 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 82**2 - 66.48**2 } }{ 2 } = 58.387 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 66.48**2+2 * 82**2 - 48**2 } }{ 2 } = 70.682 ; ;
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